Quadratic Polynomial Help?

Question

A rectangular garden has a length that is twice its width. The number of feet in the perimeter of the garden is equal to the number of square feet in its area. What are the dimensions of the garden?

Answer 1

width = x

length = 2x

 

Perimeter = Area
 
2(length) + 2(width) = length × width
 

 

Plug in the variables into the equation.

 

2(2x) + 2(x) = 2x²

 

4x + 2x = 2x²

     6x  =  2x²

 

 

Subtract 6x on both sides of equation.

 

0 = 2x² - 6x

 

 

Factor the right side of equation.

 

0 = 2x(x - 3)

 

 

Set the factors equal to zero.

 

2x = 0         and        x - 3 = 0

 

x = 0           and          x = 3

 

 

Since we cannot have a zero value for dimensions, we accept x=3.

 

 

Substitute this value of x into the dimensions.

 

width = 3 feet

length = 6 feet

 

 

Now, we can check to see if the perimeter is equal to the area.

 

Perimeter = Area

 

2(3 feet) + 2(6 feet) = 3 feet × 6 feet

 

6 feet + 12 feet = 18 square feet

 

  18 feet  =  18 square feet

 

 

The dimensions check out.

 

Answer 2

If we create two variables, one for the length (L) and the other for the width (W), we can write two equations to model this situation:

 

The length is twice the width, so  L = 2W.

 

and the perimeter is equal to the area (with different units), so   2L + 2W = LW

 

Now we can substitute the first equation into the second to get   2L + L = LW

 

Then using algebra we can derive   3L = LW,   so W = 3.   And L = 6.

 

We have a garden that is 6 long and 3 wide, so its length is twice its width.

 

And the perimeter is 3 + 6 + 3 + 6 = 18 feet, while the area is 3 x 6 = 18 square feet, so the perimeter and the area have the same number of units (one being feet, the other square feet).